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Metric projection

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In mathematics, a metric projection is a function that maps each element of a metric space to the set of points nearest to that element in some fixed sub-space.

Formal definition

Formally, let X be a metric space with distance metric d, and let M be a fixed subset of X. Then the metric projection associated with M, denoted pM, is the following set-valued function from X to M:

p M ( x ) = arg ⁡ min y ∈ M d ( x , y ) {\displaystyle p_{M}(x)=\arg \min _{y\in M}d(x,y)}

Equivalently:

p M ( x ) = { y ∈ M : d ( x , y ) ≤ d ( x , y ′ ) ∀ y ′ ∈ M } = { y ∈ M : d ( x , y ) = d ( x , M ) } {\displaystyle p_{M}(x)=\{y\in M:d(x,y)\leq d(x,y')\forall y'\in M\}=\{y\in M:d(x,y)=d(x,M)\}}

The elements in the set arg ⁡ min y ∈ M d ( x , y ) {\displaystyle \arg \min _{y\in M}d(x,y)} are also called elements of best approximation. This term comes from constrained optimization: we want to find an element nearer to x, under the constraint that the solution must be a subset of M. The function pM is also called an operator of best approximation.

Chebyshev sets

In general, pM is set-valued, as for every x, there may be many elements in M that have the same nearest distance to x. In the special case in which pM is single-valued, the set M is called a Chebyshev set. As an example, if (X,d) is a Euclidean space (R with the Euclidean distance), then a set M is a Chebyshev set if and only if it is closed and convex.

Continuity

If M is non-empty compact set, then the metric projection pM is upper semi-continuous, but might not be lower semi-continuous. But if X is a normed space and M is a finite-dimensional Chebyshev set, then pM is continuous.

Moreover, if X is a Hilbert space and M is closed and convex, then pM is Lipschitz continuous with Lipschitz constant 1.

Applications

Metric projections are used both to investigate theoretical questions in functional analysis and for practical approximation methods. They are also used in constrained optimization.

External links

References

  1. "Metric projection - Encyclopedia of Mathematics". encyclopediaofmath.org. Retrieved 2024-06-13.
  2. Deutsch, Frank (1982-12-01). "Linear selections for the metric projection". Journal of Functional Analysis. 49 (3): 269–292. doi:10.1016/0022-1236(82)90070-2. ISSN 0022-1236.
  3. "Chebyshev set - Encyclopedia of Mathematics". encyclopediaofmath.org. Retrieved 2024-06-13.
  4. Alber, Ya I. (1993-11-24), Metric and Generalized Projection Operators in Banach Spaces: Properties and Applications, arXiv:funct-an/9311001, Bibcode:1993funct.an.11001A
  5. Gafni, Eli M.; Bertsekas, Dimitri P. (November 1984). "Two-Metric Projection Methods for Constrained Optimization". SIAM Journal on Control and Optimization. 22 (6): 936–964. doi:10.1137/0322061. hdl:1721.1/2817. ISSN 0363-0129.
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